Yes, Fennec can talk to the Authenticator and ask for a "weblogin:" token for "com.google" but the Authenticator answers differently depending on who asks.
If Chrome is asking then the returned token redirects you to https://accounts.google.com/ and immediately logs you in, but when you'r Fennec then you are just redirected to https://accounts.google.com/ and have to enter username and password. Bummer.

Anyway: How about using this scheme for Mozilla services and using a Mozilla account on the device or local to the browser (Firefox Sync) if available.

get redirected to
https://accounts.firefox.com/ServiceLogin?service=bugzilla&passive=true&rm=false&continue=https://bugzilla.mozilla.org/show_bug.cgi?id=1030650
&ss=1&scc=1&ltmpl=bugzilla&emr=1

the response includes an x-auto-login HTTP header in the response

Firefox sees the x-auto-login header and- on desktop look for Firefox Sync account
use it to obtain a token from a token endpoint hosted at mozilla.org- on Android ask the AccountManager for a weblogin token for "org.mozilla".

redirect to the token (the token is an URL). In this case e.g. https://accounts.firefox.com/?t=accesstokenb64&...